Structural analysis by electric circuit analogies

STRUCTURAL ANALYSIS BY ELECTRIC CIRCUIT ANALOGIES. BY

V. BUSH, Eng.D., Massachusetts Institute of Technology.

ABSTRACT. Electric circuits m a y be constructed which obey the same laws as structures under load. Measurements on the circuits will then give the stresses, deflections, etc., of the corresponding structures. Circuits are given for determinate and indeterminate pin-connected and rigidly-connected structures in the static and t h e dynamic state. Experimental work is reported on relatively simple typical cases to show the a p p a r a t u s and methods of measurement which are applicable, and the precision obtainable.

Structures may be analyzed by three principal methods. First is the method of setting up the governing equations a n d solving them formally, in some cases by the use of approximations which greatly shorten the work. Second, mechanical models may be examined, as with a shaking-table, usually with the aid of the principle of dynamic similitude. Third, an analogous electric circuit, which is governed by the same equations as govern the structure, may be prepared and measured, and the results interpreted to give the behavior of the structure. All three of these methods will have fields of usefulness, but the last has not yet been developed to the point where its utility m a y be evaluated. Hence, in this paper, a wide range of treatment of structures by electrical analogy will be presented, with the realization that experience will be necessary to indicate parts of the method worthy of further development. The purpose of this paper is hence to develop a method of analysis of structures by means of measurements made upon analogous electrical circuits. It will be shown to be possible, by means of circuits of reasonable complexity, to treat indeterminate as well as determinate pin-connected or rigidlyconnected structures in either the static or the dynamic state. 289

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V. BusH.

[J. F. I.

Under the dynamic state can be treated impact, or generally variable loading, or the impression of forced motions such as occur in earthquakes. A t t e n t i o n will first be given to pinconnected structures, and the m e t h o d will then be extended to include rigidly-connected frame structures. T h e measurem e n t s will yield the stresses and deflections with reasonable precision. Two dimensional or three dimensional cases m a y be treated. This m e t h o d is furthered by two fundamental ideas: first, the simultaneous use of several interconnected networks to represent a single structure, or the use of several inductances or capacitances to represent a single mass point; and second, the use of ideal transformers to produce proportionality. A n y structure in the static state is governed by a set of simultaneous algebraic equations, and to obtain a solution it is simply necessary to construct and measure an electric network which is governed by this same set of equations. In this process it is convenient to produce equality by using transformers and by applying alternating potentials to the network. In the dynamic state the structure is governed by a set of linear differential equations. An electric circuit governed by the same equations m a y then be measured by oscillograph to yield the time-variation of the transient stresses and deflections of interest under variable loading. There are always a large n u m b e r of equivalent electrical networks which m a y be made to obey the same laws as those t h a t govern the performance of a given structure of moderate complexity. The simplest of these might be chosen for use. In some ways however, it is preferable to use a scheme whereby the network m a y be set up by inspection of the structure itself, and without the necessity of writing d o w n the equations involved even t h o u g h the resulting network is then somewhat more complicated. This is the procedure of this paper. T h e m e t h o d lends itself to the construction of an electrical calculating table for structural analysis. Such a table would consist of a congeries of standard electrical elements with provision for interconnecting t h e m rapidly in proper m a n n e r by direct comparison w i t h the interconnection of the elements of the structure under analysis, and with provision for accurate m e a s u r e m e n t of those electrical quantities

Mar., I934.]

STRUCTURAL ANALYSIS BY ANALOGIES.

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which represent the desired stresses and deflections in various members. This procedure is similar to that used by electrical engineers for the study of the behavior of complicated electrical power systems. An electrical replica of the system under study is set up and measured in the laboratory. The most complete representation is obtained by using an alternating current network analyzer. 1 It is the purpose of the present paper to develop a corresponding method whereby the analysis of complicated structures m a y be carried through in cases of such complexity that the usual methods of analysis become prohibitively laborious. The corresponding method for power-system analysis required m a n y years of development, and a considerable cost, in the construction of calculating-tables and network-analyzers before becoming generally useful; but at the present time important studies of power systems m a y be rapidly and accurately carried out in the laboratory in cases where the complexity involved would render the analytical solutions entirely impractical from the standpoint of cost and time required. It is hoped that a development of similar electrical methods for structural analysis may be developed to the point where studies may be undertaken at justifiable cost of complex indeterminate structures under both steady and variable load conditions, with a considerable reduction in time and cost, thus making possible studies which are now too laborious to be economically justifiable. The basic idea of the treatment of mechanical systems by means of measurements on analogous electrical circuits appears in several places in the literature. 2 The general scheme is to produce an electrical network governed by the same set of differential or algebraic equations as the mechanical systems, and subject to the same system of constraints, 1 Travers and Parker, "An Altelnating Current Calculating Board," Electrical Journal, May 19,3o. Hazen, Schurig and Gardner, "The M.I.T. Network Analyzer," Trans. A.I.E.E., 49, No. 3, July i93o , pp. Iro2-III3. For example: C. A. Nickel "Oscillographic Solution of Eleetromechanical Systems," A.I.E.E. Transactions, 44, 844 (I925); H. If. Skilling, "An Electric Analogue of Friction for Solution of Mechanical Systems," Trans. A.I.E.E., Sept. 193I, p. 1155; Chapter I6 of "Operational Circuit Analysis," J. Wiley & Sons Co., by V. Bush.

292

V.

BUSH.

[J. F.

I.

to set up the analogy between mechanical and electrical quantities, with convenient conversion factors, and then to measure the electrical quantities in order to obtain the value or time variation of the corresponding mechanical quantities. For static problems of determinate or indeterminate structures the work of Mallock is outstanding? This treats the solution of simultaneous linear algebraic equations with real coefficients by electrical measuring methods in a general manner, and hence provides a way of solving static problems of structures. The appearance of Mallock's excellent treatment has made it possible to greatly shorten the present paper, although the consideration of static cases is still included largely because the method of approach is here directly from the structure to the network, rather than by first setting up the governing equations, and also because of certain differences of detailed treatment. Considerable use will be made below of an " i d e a l " transformer. This is a transformer in which, under the conditions of use and to the degree of approximation necessary for accuracy of results, the primary and secondary currents or voltages are always in a fixed constant ratio. Such a transformer will ordinarily be built with a nickel-iron alloy core and closely coupled windings. In the case where primary and secondary currents are to be always in a certain ratio, the same conditions hold in the transformer design as hold f o r current transformers used in measurement work. The flux density should be low, and the primary turns sufficient so that the exciting current is negligible compared to the work current. Frequency should be high in order to reduce size, but low enough so that the currents flowing through distributed capacities will also be negligible compared to the work current. In some cases the transformer leakage reactance should be small. A similar set of conditions can readily be stated when it is desired to maintain a fixed ratio of voltages. A transformer of this sort, inserted in an electrical network, forces substantial equality or proportionality between two electrical quantities. Another important artifice used below is the treatment of a R. R. M. Mallock, " A n Electrical Calculating Machine," Proc. Royal Soc., I4o, No. A84I, May 3, I933, p. 457.

Mar., 1934.] STRUCTURAL ANALYSIS BY ANALOGIES.

293

stresses and reactions in each dimension separately. A single mechanical system thus becomes represented by two electrical networks, in the case of a structural problem in a plane, or by three networks in the case of a three-dimensional problem. These networks are interconnected, and together they represent a ~ingle structure. When the inertia of a mass point is to be represented, the equivalent electrical element appears in each of the networks. When a diagonal structural member appears, its reaction is introduced into the several networks by transformers of correct ratio. The simplest problem to be treated is that of the static stresses in a determinate structure such as a truss in a single plane, and having pin joints. This will be discussed, not because of the value of the results, for those can be obtained very easily by graphical methods, but in order to illustrate principles which are used in more complicated problems. An example will be treated from which the general application will be evident. Consider the truss of Fig. I, with a s;ingle load at its center. The procedure of setting up the electrical networks is as follows: make two diagrams, one for horizontal and the other for vertical stresses, locating points in each by correspondingly numbered panel points. In the horizontal network connect together each pair of panel points which in the structure are connected by a horizontal member. Similarly, in the vertical network, connect each pair which in the structure are connected by a vertical member. In both networks connect each pair which in the structure are connected by an inclined member, but interpose in these connections in each case a transformer, the primary of the transformer being inserted in the connection of the horizontal network and the secondary in the connection of the vertical network. Let the ratio of each tarnsformer be such that the current in the horizontal network connection bears the same relation to the current in the vertical network connection, that, in the structure, the horizontal stress in the member bears to the vertical stress. In other words, let the ratio of the transformer, measured as the ratio of secondary to primary current, equal the tangent of the angle of inclination of the member with the horizontal. At the point where the vertical load is applied, in this case

294

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panel-point 4, bring out a lead from the vertical diagram, and insert in this lead a source of alternating current A. The return leads are connected to the diagram at the points of reaction of the supports. Now the method of using the network is simple. Adjust the current at A to any convenient value I, using any freFIG. I . 8

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quency suitable for proper transformer behavior, and let this current represent the load W. To get the stress in any horizontal or vertical member read the current in the corresponding connection and multiply by W/I. To get the stress in an inclined member read the current in either vertical or horizontal connection and multiply by W/I and by the cosecant or secant respectively of the angle of inclination.

Mar., ~934.] STRUCTURALANALYSIS BY ANAI,OGIES.

295

The proof of the procedure so far m a y be very brief. At a panel point of the horizontal network Kirchhoff's law states that the sum of the currents in the connections leading to that point is zero. The corresponding s t a t e m e n t for the structure is that, at a panel point, the sum of the horizontal components of the stresses in the connecting members is zero. Similarly, for the vertical network aad vertical stresses. There is hence a one-to-one correspondence between currents in the horizontal network and horizontal stresses in the structure, and between currents in the vertical network and vertical stresses in the structure. Exactly the same laws apply to currents in one case as apply to stresses in the other, and in each case these laws are sufficient to fix the distribution. Hence we have an accurate analogy. The extension to more complicated problems of this general nature is now simple. If there are several loads, then there will be several sources of alternating current, and each current must be adjusted until its magnitude represents to scale the corresponding load. If there is no symmetry, then a separate source must be used to adjust the reactions. All the sources must be adjusted so that the currents supplied are strictly in phase. If there are horizontal loads, or horizontal components of loads, t h e y are handled by sources connected to the horizontal network. If the problem is a three dimensional one, three networks are necessary. All panel points appear in each network diagram, although, since there is no need that they be placed in their correct geometrical relationship, t h e y can all appear in a plane for convenience. Connections are made as before. A member located in a principal plane is treated exactly as above. When a member is inclined with respect to all three axes the corresponding electrical connections are interrelated by transformers in such manner as to preserve the correct relationship between all three components of stress, two transformers being necessary for this purpose. The rest of the procedure is self-evident. In this type of problem the only circuit elements utilized are current transformers. The only voltages appearing are those due to the flow of currents through the leakage reactances and internal resistances, and these should be small and

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immaterial. The single criterion for accuracy is that the exciting currents of the transformers, sufficient to produce these small voltages, shall be negligible compared to the currents measured. In a problem of the sort illustrated in Fig. I, s y m m e t r y may be made use of, and only half of the circuit actually set up. The inclinations of the inclined members are all equal, and hence all the transformers m a y be of the same ratio. Moreover, by using different conversion factors for the two networks, transformers of unity ratio may be utilized. In other words the factor due to the angle of inclination may be introduced into the computations rather than into the transformer ratio. Some of these devices m a y be useful in connection with later and more complex problems. There will next be considered the much more important problem of the indeterminate structure, still with pin connections, and at present with static loads. Again we will consider an explicit example, from which it will easily be possible to generalize. Consider the simple indeterminate structure of Fig. 2. We now have to consider the compliance of the members and we will call R m n the compliance of the member connecting points m and n. This is the change in length of this member produced by unit axial stress. As before we make two diagrams, one horizontal and the other vertical, and number the panel points. In the horizontal network connect all pairs of points which in the structure are connected by a horizontal member, but into this connection insert a resistance proportional to the compliance of the member. In the figure this resistance is shown labelled as equal to the compliance, but any convenient conversion factor m a y be used and employed in the final comparisons. Proceed similarly in the vertical network. Where inclined members appear, place the corresponding resistance in both networks, and interconnect the two networks at this point by means of a transformer, the ratio of which is chosen just as outlined above for determinate structures. Load is applied as before. In the example under consideration a horizontal load is shown at points 3 and 4. This is represented by a source of alternating current connected as

Mar., I934.] STRUCTURAL ANALYSIS BY ANALOGIES. FIG. 2. 2

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shown. Adjust this current to a convenient value, and obtain the stresses in members by measuring the corresponding currents and using conversion factors as outlined above. The deflections, t h a t is the change in length of members, m a y now be obtained by measuring the voltage across the corresponding resistances and using the conversion factor setup when these resistances were chosen. M e a s u r e m e n t s on inclined members in the two diagrams give horizontal and vertical deflections respectively, and to obtain the total length change we m a y take the square root of the s u m of the squares of these values. T h e analogy between currents and stresses is set up exactly as before. T h e s t a t e m e n t that, at a panel point in the horizontal network, the s u m of the currents is zero, is exactly analogous to the s t a t e m e n t t h a t at a panel point in the structure the s u m of the horizontal forces is zero, and so on. The analogy between voltages and deflections comes about as follows: Across a resistance corresponding to a horizontal m e m b e r appears a voltage which is equal to the product of the resistance and the current. In a horizontal m e m b e r of the structure appears a change of length which is equal to the product of the stress in the m e m b e r and its compliance. Similarly for vertical members. Across the resistance in the horizontal network corresponding to an inclined m e m b e r appears a voltage which is equal to the product of the current and the resistance. In an inclined m e m b e r of the structure appears a horizontal c o m p o n e n t of change of length which is equal to the product of the horizontal c o m p o n e n t of stress and the compliance of the member. Similarly in the vertical network. Moreover, we have the following relationships: The rate of power dissipation in a network element is P R . The total power dissipation is E I 2 R taken over all the elements. The distribution of currents in the network will be such as to make this a m i n i m u m , t h a t is, ~EI2R = o.

T h e potential energy stored in a m e m b e r of the structure is P 2 R / 2 , where P is the stress and R is now the compliance. T h e total potential energy stored is Y.P~R/2. T h e distribution of stresses in the structure will be such as to render this

Mar., ~934.] STRUCTURAL ANALYSIS BY 2~tNALOGIES.

299

total stored potential energy a minimum, that is, ~P~R

= o.

Thus the distribution of currents in the network is controlled by exactly the same law as is the distribution of stresses in the structure. Note must be taken, however, of the fact that we have two networks and one structure, and that the network currents represent stress components. ] ' h a t the relationship still holds under these conditions is shown as follows: The minimum dissipation relationship holds for either network alone, giving 6~IJR

= o

6ZIu2R

= o,

and where Ix and Iy are currents in corresponding elements of the horizontal and vertical networks. To every term in the first equation corresponds a term in the second which involves the same value of resistance, with the remark t h a t where vertical or horizontal members are involved one of the component currents is zero. We can therefore add, and obtain ~(.[

2 @ L2)R

= o.

But I J + Iu 2 is the q u a n t i t y which we consider analogous to p2. Our analogy is hence complete. In order to examine the accuracy obtainable with apparatus of the usual type, a problem involving the static stresses in an indeterminate pin-connected truss was worked out analytically and experimentally for comparison. The problem investigated is shown in Fig. 3, which gives the essential mechanical dimensions. ]'here are two indeterminate panels, and the slope of all slanting members is 45 ° . One to one current transformers are therefore required. The resistances in ohms were made equal to the ratio of length in feet to area in square inches of the members. The transformers used were of the usual commercial type of instrument transformers, rated 5 amperes, 8o watts. Tested for ratio, the error with 3.2 ohms load was less than o.I per cent. from o.5 to 5.o amps. With 13 ohms load the error was less than VOL. 217, NO. I299--I9

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2 per cent. at 0.5 amps., and decreased with increase of current. The use of these transformers for this problem was therefore considered satisfactory. FIG.

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3.

b

c

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Length/Area, Member. P1 TI T~ Px P2

ft./sq, in.

.......................... .......................... .......................... .......................... ..........................

I.OI 2.I43 3.030 3.o3o 3.333 PI .......................... 3.535 Pr . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . S 3 5 P~ .......................... 3.333 T~ ......................... 3.030 T1o . . . . . . . . . . . . . . . . . . . . . . . . . . 2.I43 P~ .......................... i.OlO a ............................ 769 b ............................ 769 c ............................ 769 i ........................... 1.364 2 ........................... L364 3 ............................ 937 XO . . . . . . . . . . . . . . . . . . . . . . . . . . 1.364 20 .......................... x.364

1o Length, Feet. 42.4 3o 42.4 42.4 3o 42.4 42.4 30 42.4 3o 42.4 3o 3o 3o 3o 30 3o 3o 30

Pin connected truss problem.

The circuit used to represent the two indeterminate panels is shown in Fig. 4. The primaries and secondaries of the current transformers are indicated by circles, and the cores by d o t t e d lines, for the sake of clarity. The external loads on t h e structures are applied through the reactions labelled R - R in the figure. The results are indicated in Table I, which compares the calculated stresses with the observed. In setting u p this prob!em it was found that the transformers used had a sufficiently large leakage reactance to affect the results, when pure resistances were used to represent the compliance of members. It is not at all necessary that there be pure resistances; in fact, any sort of impedance may be used, provided only that the phase angle of each impedance be the same. Accordingly, each branch was made up of a resistance plus an inductance, and the latter were chosen so that,

Mar., I934.]

STRUCTURAL

ANALYSIS FIG. 4"

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Circuit for pin-connected truss problem. Resistances equal (Length/area) ohms. Reactances included in all members to make ratio of resistance to reactance constant; leakage reactances of transformers the determining factor. Circuits coupled by transformers indicated by dotted lines.

together with the leakage inductance of the transformers, they produced equal impedance angles. This procedure is, of course, unduly laborious for rapid work. Accordingly, in any permanent arrangement, the transformers should be so designed that their leakage reactances are negligible, or else condensers should be permanently connected to each transformer so as to compensate at all times for the effect of leakage reactance.* Mallock, loc. cit., has developed an ingenious a n d satisfactory m e t h o d for avoiding difficulty due to imperfect t r a n s f o r m e r action which m a y be extended for use in cases of this sort.

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TABLE I,

Loading and Stresses in Thousands of Pounds in the Two Indeterminate Panels of Fig. z.

Loading.

Member. a

b 2

3

TI

P2

300 lbs. a t A.

Calc.

Meas.

247. 141 • 173. 159. 233. 28.2

247. 141. 174. 158 . 229. 28.9 20.9 lO.O6 95.9 57.0 30.2

21.2

T2

P. Pz Pr

9.8 94.7 54.9 30.0

300 lbs. a t B. Per Cent. Error.

o.5 0.6 1-7 2.5 1.3 3.0 i .2 3.5 0.6

Calc.

I5o a t each of C and D. Per Cent.

Calc.

Meas.

o. 4

147.7 232.2

229. 5

0.5 1.8 1.2 0.5 0.5

127.3 217.8 37.3 15.o 37.8 74.6

67.4

I.O

52.7

70.8 98.8

0.3 0.5

73.8 53.5

Meas.

312,

312.5

31o.3 228, 29o. 48.2 97.8 70.3 186. 68.2 7o.6 99.4

3o9. 228. 288.5 49.o 99.0 70.0 i87 .

Error.

I42.9 I26.8 215.2 37.2 14.6 38.o 74.6 ' 51.6 73-7 " 53.6,

Per Cent. Error. O.I 1.0

o.5 1.2 0.2

2.7 O. 5 -2.0

---

This problem was worked out using ordinary meters for measurement. As the insertion of the a m m e t e r appreciably affected the circuit, a compensation for the a m m e t e r resistance was made by changing the corresponding branch resistance each time the meter was inserted in such a way as to hold the total resistance constant. Again this is too slow a procedure for permanent use, and a thermionic-tube a m m e t e r constructed so t h a t it could be inserted at will in any part of the network without disturbing its performance would be much better. Such a meter is readily designed and was used in later work. The results check computation sufficiently closely for m a n y engineering purposes. It will be observed t h a t the percentage error tends to be larger in members with low stresses. The extension to more complicated, pin-connected, indeterminate structures can now be made without difficulty. Each inclined member requires two adjustable resistances and one transformer in order to represent it, and each other member requires one resistance. Experience with the network analyzer gives a basis for estimation of the time necessary to make solutions, provided sufficient suitable apparatus were conveniently available for the purpose. Consider as an example a structure, of the type now under consideration,

Mar., 1934.] STRUCTURAL ANALYSIS BY ANALOGIES.

3o3

having IOO members. To set up and adjust the corresponding network would require perhaps two hours. Any usual system of loads could then be applied in a few minutes. The stresses all over the structure could then be read and recorded in an hour. The limit to the complexity of structure that can be treated is set only by the number of electrical elements available. So far only the static case of the pin-connected structure has been considered. We now turn to the more complicated problem of the static stresses in a rigidly connected structure. In a structure of this sort we have to consider moments and shears, and angles of members become equally important with deflections. We will treat only cases in which all members are vertical or horizontal, and in which the longitudinal compliance of members is negligible by comparison with their flexibility as beams. When there are diagonal braces it is usually true that longitudinal rigidity will control, and that moments can then be neglected, so t h a t we revert to the preceding cases. In some few structures there are diagonal members and yet the reactions due to bending and longitudinal strain are of the same order of magnitude and an extension to these cases can be made. In the present paper the t r e a t m e n t will be limited to the structure, similar to that of a frame of a building, without diagonal members. A single member will first be considered, and then the interconnection of members. A single member is characterized by the shear in the member, the moments at the two ends, the angles of the two ends, and the lateral deflection of the two ends. Such a member is represented by the network of Fig. 5. All transformers are current transformers. The analogy is always with current; t h a t is, with different conversion factors in each case, current gives shear, moment, angle or deflection. Let moments acting on a member be positive if acting in a clockwise direction; positive angle be in a clockwise direction; positive relative deflection in a member correspond to a clockwise rotation of the line joining the ends of the member; and positive shear produce positive deflection. Then positive shear m a y be represented by current flowing into the A end of the member and out of the B end. Positiw~ moments at either

304

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FI~. 5.

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0

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0

0

0

0

0

Mar., I934.] STRUCTURAL ANALYSIS BY ANALOGIES.

305

end are represented by current flowing into the member. Positive angle is given by current flowing in at the A end and out at the B end. Current flowing in at the A end and out at the B end represents the absolute deflection of the A and B ends respectively, so t h a t the relative deflection in the m e m b e r is represented by the difference between these two currents. If the weight of the members be neglected and the loads including wind loading be treated as if applied at panel points only, then the equations of an individual m e m b e r m a y be written : MA + M s + S L = o OA = 08 + M A ( L / E I ) + S(L~'/2EI) ,5A -- ,58 = 08 + MA(L2/2EI) + S(L3/3EI),

where S is the shear in the member, Ma the m o m e n t acting on the A end of the member, M8 the m o m e n t acting on the B end, 0a and 0s the angle of rotation of the A and B ends respectively, and AA AB the relative deflection in the member. T h e transformer connections indicated in Fig. 5 impose these relations. T h e transformer ratios are determined by the length and stiffness of the m e m b e r and the scale factors chosen. Actually, as far as a single m e m b e r is concerned, the the transformer ratios m a y all be I : I except for a single unit if scale factors are properly selected. It remains to interconnect such units in order to represent a structure. The terminals of Fig. 5 are for use in connecting to adjacent units, G being a c o m m o n terminal. Thus, if terminals m a r k e d G and 08 are connected to terminals G' and 0a' of an adjacent network, the value of the current leaving 0B represents the angle of the m e m b e r here represented at end B, and the interconnection forces this to be equal to the angle of the adjacent m e m b e r at end A, and so on. Consider four structural members meeti,ag at a point, as shown in Fig. 6. In the diagram of the network of this figure the unit shown in Fig. 5 is indicated simply by the symbol shown at the b o t t o m of t h a t figure, the unlabelled binding posts corresponding to those of the full diagram above. The interconnections are such as to preserve the following relations between the members meeting at the panel point. First the sum of the m o m e n t s is m a d e zero, by joining the four corre-

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Mar., I934.] STRUCTURAL ANALYSIS BY ANALOGIES.

3o7

sponding circuits. Second, by the use of a I : I transformer, the angles of all members at the panel poin.t are m a d e equal. This transformer is indicated symbolically by a circle, the actual connection being as shown at the right. Third, the deflections of adjoining ends of vertical members are m a d e equal by series connection. T h e same is done for adjoining horizontal members. T h e two pairs are not equated, however, for the deflections are in each case lateral, and hence at right angles and independent. Finally, the shears are interconnected in a m a n n e r which will be clear in a complete structure representation. Let us now consider the complete structure of Fig. 7. This shows a frame of nine panels, or twenty-one members, acted upon by horizontal forces, and connected at the base to a foundation. The problem m a y be that of wind pressures on a steel frame building. In the corresponding network shown in Fig. 8 there is a unit, such as shown in Fig. 5, for each member. T h e complete network is complicated, but it m u s t be remembered t h a t the formal t r e a t m e n t of this problem would involve the solution of a large n u m b e r of algebraic equations if no algebraic reductions were employed so t h a t the complexity of the network is forced by the inherent difficulty of the problem being attacked. As a m a t t e r of fact the network can be simplified by taking advantage of symmetry, thus making it necessary to represent only half the structure, and equations m a y be greatly combined before proceeding to the electrical set-up, but in the present explanation it is clearer to use the full diagram. Note first t h a t wherever members meet at a panel point the sum of the m o m e n t s is made zero by complete interconnection. Second, wherever members meet, the angles are made equal, by series connection in the case of two members, and by transformer in the case of three and four. Third, where two vertical members meet, the lateral deflections are made equal by series connection. However, the lateral deflections of horizontal members m u s t be zero, so the corresponding circuits of members I~) are left open. The lateral deflections of all vertical members are m a d e equal at each floor level by transformer connections. Also the deflection, as well as the angles, of the lower ends of members IO, 13, 16,

308

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19, are held at zero b y the are also left open. Finally, • applied loads are represented n a t i n g c u r r e n t F1, F2 and ?'3.

[J. F. I.

foundation, so these terminals let us trace the shears. T h e b y adjustable sources of alterThese supply currents are held

FIG.

7.

I

I i I

2

I

I

I

[ l

I l

...0_

I B

B

A

Continuous franle building. For each member: E = 3 X IO r l b s . / i n 3 I = IOO0 in. 4 l = I00 inches.

strictly in phase. Similarly, the currents which appear at R1, R2, R3, a n d R4 represent t h e reactions which a p p e a r a t the foundation. T h e load FI is equal to the s u m of the shears in m e m b e r s 12, I5, I8, 2I. T h e five corresponding connections

Mar., 1934.] STRUCTURAL ANALYSIS BY ANALOGIES.

309

are brought to a k n o t p o i n t to force this condition. Similarly for reaction R1 and the shears in I, 4, 7 and also for R4. Next we have the fact t h a t the s u m of shears in I I, I4, 17, 20, minus the shears in 12, I5, 18, 2I equals F2. This is forced by the interconnection shown, in which the connections representing all these shears and t h a t for F2 are brought to a knot point. The other load F3, and the reactions R2 and R3 are treated in similar manner. Care m u s t be taken, of course, to maintain algebraic signs correct in m a k i n g the connections. T h e network is now completely constrained, and the constraints correspond to those in the structure. Measurem e n t s will now give shears, moments, angles, and deflections t h r o u g h o u t the structure. Moreover, any other system of loading can be examined without m a k i n g new network interconnections by simply altering the currents representing the loads. If the loads are not in the plane shown they can be split into components which can be applied to the networks representing two sections of the structure, and they will act independently. Usually only one c o m p o n e n t requires examination. This t y p e of s t r u c t u r e requires a large n u m b e r of currenttransformers for its representation. It was hence i m p o r t a n t to determine whether reasonable accuracy could be obtained with transformers of an inexpensive sort. T h e effect of transformer errors on the final result cannot be predicted in any simple way by computation, so experimentation in a typical problem was resorted to in order to obtain an indication of w h a t might reasonably be expected. T h e moments, shears, angles, and deflections resulting from wind load on a simple frame structure were determined. T h e structure chosen was t h a t of Fig. 7. T h e loading of IOOO lbs. applied horizontally at each floor level was assumed. Symm e t r y was taken a d v a n t a g e of in order to reduce the network complication. Members representing half the center bents m a y be used, and the conditions at the center line imposed. This was simplified still further by recognizing the fact t h a t so far as the remainder of the structure is concerned, only the relation between m o m e n t s and angular rotation at the ends of these bents need be taken into account. A single transformer is sufficient to impose this condition. Between members, the

31o

V. B u s H .

[J. F. I.

FIG. 8.

O

I

2

3

4

S

6

7

8

9

__

©

7

i

Mar., 1934.] STRUCTURAL ANALYSIS BY ANALOGIE.q.

311

b o u n d a r y conditions are imposed by the proper connections, as treated above. The resulting n e t w o r k is shown in Fig. 9. T h e design of the transformers used was based upon the d a t a for H y p e r n i k given by Yensen? The data on various thicknesses of laminations were extrapolated, which m a y account for some of the difficulty later encountered. The loss c o m p o n e n t of exciting current at sixty cycles per second was considered negligible for the thickness of laminations used (o.oo5"). Subsequent tests indicated t h a t this was not the case, although the fault m a y have been with the t r e a t m e n t given the laminations. However, fairly satisfactory transformers were obtained. T h e cores were made up of continuous annular laminations 2 1-!6"outside d i a m e t e r , ,~43_,, inside diameter, .oo5" thick, annealed and lacquered. Forty-five laminations were used, giving a cross section of .2 4 square centimeter. The basic winding was 222 turns of No. 19 B. & S., enameled and cotton covered, doubled, for both primary and secondary. The coil was p u t on by hand, carrying the four strands through together, both to insure a one-one ratio and to produce a m i n i m u m of leakage reactance. Testing the ratio of transformation of various units and under various conditions gave an average error of about 1.5 per cent. m a x i m u m for a load equivalent to five times the resistance of a single transformer, over the range of currents for which the transformers were designed, and at 60 cycles per second. This is none too good and can certainly be improved upon without increasing costs in later designing. T h e object of the tests, however, was to determine the overall errors caused by imperfect transformer behavior, and it was hence an advantage, rather than otherwise, to utilize transformers somewhat less perfect than could be produced at low cost. A set of experiments at 6o cycles per second soon showed t h a t the transformer behavior noted above is not nearly good enough. The actual network is so complicated t h a t it is not easy to visualize the conditions under which individual trans5 T. D. Yensen, "Permeability of Hypernik Reaches 167,ooo" ; El. Jl., 28, No. 6, June I93I.

312

V. BusH. FIG. 9.

[.L F. I.

Mar.,

I934.]

STRUCTURAL

ANALYSIS

BY

313

ANALOGIES.

TABLE II.

Results, Continuous Frame Structure. Value.

Per Cent. Error.

Item. Calculated.

SAIo SBIo SA2I SB2I SA32 SB32 S3AB S2AB SIAB

........ ........ ........ ........ ........ ........ ........ ........ ........

MIAB MAI2 MAIo M2AB MA23 MA2I MA32 M3AB M3BB M3BA MB32 M2BB MB2I MB23 M2BA MBIo MBI2 MIBB MIBA

....... ....... ........ ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... .......

6 6 7 lb~. 833 374 626 183 305 219 525

774 4 1 5 N lO 2 i n . lbs. I59 255 278 64 214 119 I19 84 99

oA3. . . . . . . . . . OB3 . . . . . . . . . .

655 lbs. 835 366

1.9 0.2 --

2.2

625

I65 336

--IO

2IO

--

490 718

--

4.i 6.7

--

7.2

IO

3 9 0 N IO 2 in. l b s . I45 234 258 45 210

--

IO7 Io7 90 94 205 242 331 131

--IO

--

247

228

--

8

366 299 305 360

360 300 336 328

--

2

182

218 328 I37

Oi ........... D2 ........... D3 ........... 0AI .......... 0A2 ..........

Measured.

. 0 0 3 1 6 in. .OO319 .00172 261 X IO - r r a d i a n I7O

72 46

OB2 . . . . . . . . . .

I2I

0BI ..........

z69

--

6

--

9

--

8

--

7

--3o 2

--IO

7 -

5

I3 II I 4

IO

--9 . o o 3 1 3 in. .00324 .00167

2 6 7 M IO -T r a d i a n I66 67 46.5 I27 176

--

I

1.5 --

3

2.2 --

2. 4

--

7

I

5 4

formers operate, b u t in s o m e cases t h e y are in effect bucking each other. T h e net result is that, in order to obtain good overall precision, the individual t r a n s f o r m e r behavior m u s t be v e r y good indeed. In order to m a k e the next series of tests, however, it was not necessary to build n e w transformers. T e s t s at 5oo cycles

3I 4

V. BusH.

[J. F. I.

per second showed a greatly improved performance of the transformers already available, the error in ratio being about one-third of w h a t it was at 6o cycles per second. The relations of individual members, when separately examined, checked within one per cent. Moreover, with the precautions noted below, the tests on the entire network now yielded the results of Table II, which are probably sufficiently good for practical purposes. It should be noted t h a t the accuracy is best on the numbers having large stresses. Very large per centage errors go with small values. Thus, the smallest m o m e n t is 3 ° per cent. in error. It thus appears t h a t a good indication has been obtained of the transformer performance necessary, and this performance can readily be obtained in later designs at either 6o or 5oo cycles per second and with small units of low cost. This can be d o n e - - i n fact, a p p a r e n t l y with a considerable m a r g i n - - b y making sure t h a t losses are as low as obtainable by use of nickel-iron magnetic materials, and by either designing for low total excitation or by cancelling the magnetizing c o m p o n e n t by the use of condensers. In the tests a n u m b e r of precautions had to be taken to ensure accuracy. In order to avoid u n d u l y disturbing the network upon the insertion of the ammeter, a low resistance s h u n t was utilized with a thermionic tube amplifier. This is a satisfactory procedure when provision is made for frequent calibration. Since the resistances everywhere in the network m u s t be low, great care had to be taken to ensure low resistance joints and connections. In any p e r m a n e n t arrangement, this point will require close attention. In a flexible network, arranged to be readily connected at will to represent various structures, it will be advisable to provide means whereby the ratios of transformers m a y be rapidly checked in service. This is because of the fact t h a t it is almost impossible, by a mere inspection of the circuit diagram, to determine whether the conditions of operation of every transformer are within the range for which it was designed. ~ Some, at least, of the transformers should be provided with taps so t h a t the ratio of current m a y be readily Mallock, loc. cit., developed a method for a u t o m a t i c correction of potential transformers. This can be altered to apply to current-transformers as here used.

Mar., I934.]

~TRUCTURAL ANALYSIS BY ANALOGIES.

315

corrected if it departs seriously from normal. This procedure was carried through somewhat laboriously in the absence of special provisions, in the tests here reported. In the case of three key transformers, the ratio was found to depart notably, and was corrected by adding one or two turns. With complete e q u i p m e n t this procedure will occupy only a short time and will be a necessary precaution. This completes the t r e a t m e n t of structures in the static state. The sets of equations thus far involved have been algebraic. It is evident t h a t the network m e t h o d s employed m a y also be used for evaluating determinants. For rapid use in structural engineering, however, it appears to be preferable to adhere to the procedure of passing directly from the structure to its corresponding network. This eliminates a considerable a m o u n t of algebraic work and avoids some danger of error. This is accomplished at the expense of involving rather complicated networks, but as the units employed are inexpensive this does not appear to be a serious limitation. There is no reason except complexity why the m e t h o d cannot be extended to continuous structures, which m a y be essentially three-dimensional. In the analysis of such a structure as a concrete dam, the accuracy obtainable would depend upon the n u m b e r of units utilized and the extent to which transformer action approached the ideal. The procedure would be to arbitrarily divide the structure into units, not necessarily cubical, and to represent the interaction of each with all its neighbors in m u c h the same way t h a t has been described for the members of a frame structure. The complication would be very large if the size of the units were taken reasonably small. Yet, in view of the serious difficulty involved in analysis of this sort of structure by any means, it is desirable t h a t the m e t h o d here outlined be further investigated in this connection t h a n has been possible during the s t u d y which has led up to the present paper. T h e next problem is t h a t of the pin-connected structure under transient loading. T h e governing laws are now expressible by differential equations, and it becomes necessary to represent the masses present in the structure as well as the c o m p l i a n c e of members. The frictional d a m p i n g m a y also be represented, although in a structure it is usually small, and VOL. 217, NO. 1299--20

IJ. F.

V. BUSH.

316

FIG. IO.

2

I

3

•

4

LI

L2

HOR. D

L3

L4

H

C231 LI

E

L2

VERT.

L3

L4

M a r . , 1934. ]

STRUCTURAL

ANALYSIS

BY

ANALOGIES.

317

often highly inestimable. The masses will be considered concentrated at the panel points. There are two electrical analogues which m a y be used and the more readily visualized of these will be considered first. In this, masses will be represented by inductances, and the compliance of m e m b e r s by condensers. Mechanical friction, if taken into account, is represented by electrical resistance. A very simple structure is shown, for illustration, in Fig. IO. We m a k e two networks as before. Each mass is represented in both networks by an inductance L , which is proportional to Mn, the mass at the corresponding panel point in the structure. A n y desired proportionality factor m a y be used. In the horizontal network we connect in series any two masses which in the structure are connected by a horizontal member, and similarly in the vertical network. Across this interconnection we connect a condenser Cm, representing the compliance of the connecting member. Again the proportionality factor between capacitance and compliance m a y be chosen at will. For an inclined m e m b e r we interconnect corresponding masses in both networks, using transformers of ratios chosen as indicated above. Across both of these interconnections we connect a single condenser representing, to the same scale as for the condensers above, the compliance of the inclined member. The analogy is as follows: In the Structure. Mass ........................ Resilience ................... Velocity ..................... Displacement ................ Stress .......................

M S V l F

In the Network. Inductance. L Capacitance C Current .... I in a c o i l Charge ..... (2 t h r o u g h a coil Voltage .... Ir on a condenser

Horizontal velocities, displacements, and stresses appear represented in a horizontal network, and similarly for the vertical network. Notice t h a t the interconnection is such t h a t the charge appearing on a condenser representing the compliance of a horizontal m e m b e r is proportional to the difference in the horizontal displacements of the panel points to which it is connected, and similarly for the vertical members. For an inclined m e m b e r the relationship is as follows: Consider the

318

V. BUSH.

[J. F. I.

transformer el, to be explicit. The charge passing through the primary is proportional to the horizontal displacement of point 2 of the network. The charge passing through, the secondary is hence, in accordance with the transformer ratio, proportional to the change in length of member 2- 3 produced by such a displacement acting alone. Thus the charge passing the leads at point D is proportional to the change in length produced by the difference of the horizontal displacement of points 2 and 3. Similarly the charge passing E is proportional to the change in length produced by vertical displacements. The charge appearing on condenser C~.ais hence proportional to the net change in length of member 2-3. All static relationships are hence maintained, for the voltage across a condenser is equal to the quotient of charge and capacity, and the stress appearing in a member is equal to the quotient of change of length and resiliency. It remains only to examine dynamic relationships, which is a simple matter. The voltage across any given condenser is applied to both of the inductances to which it is connected and tends to produce a change of current in them. The stress in a given member acts on both masses with which it connects, aad tends to accelerate them. We have also

LdI= V dt and

dV M~[---F. The same set of differential equations hence governs both the behavior of the network and the motion of the structure. The analogy is hence complete. The network may be simplified by altering scale factors in order to obtain unity-ratio transformers, and by omitting transformers in cases where it is not necessary to retain independence of circuits. Thus, if the structure of Fig. IO is square, the network can be reduced to the simple form shown in Fig. I I. Since stresses are represented by voltages the sudden application of a load is represented by the sudden application of a constant potential. Thus the sudden application of equal

Mar., I934.] STRUCTURAL ANALYSIS BY ANALOGIES.

319

opposing horizontal forces to points 3 and 4 of the structure would be represented by the sudden connection of a battery in series with the circuit at point H of Fig. IO or Fig. I I. Measurements are now m a d e by oscillograph, preferably by aid of a thermionic tube amplifier in order not to disturb the network. The oseillograms give the time variation of velocity, and hence, by integration, of displacement, or of stress. The proportionality factor between the time scale for the oscillogram and for the structure is readily evaluated. The conditions for proper transformer action are simple, as the effect of FIG.

I I.

L!

L2 I

HOR.

L3 d

I

~2

Lt

C 22

VERT. 13

L3

L4

leakage reactance m a y be offset by adjusting external inductances. Cases will often occur where certain panel points are fixed, as when t h e y are anchored to the foundation. There can then be no motion of these points under ordinary conditions and this is provided for by opening the circuit of the corresponding inductances. This simplifies the network at these points. Thus, if points 3 and 4 of Fig. Io are anchored to the foundation, the circuit reduces to t h a t of Fig. 12. In this diagram is shown an alternator at K, representing a sinusoidally variable

320

V. Busi-i.

[J. F. I.

force applied horizontally at mass I, such as might be due to a varying wind. The vertical motion of mass z is omitted in the circuit as there is no cause present to produce such a motion. Now generally variable forces m a y be applied just as well as simple ones. For example, we have the important problem of the stresses produced when a train runs over a bridge. To simulate this problem we have to apply to a number of panel points a load which varies in a known manner with the time. It is hence necessary to produce and apply a set of voltages having a prescribed time variation: There are numerous ways of accomplishing this. One way, perhaps not the simplest, is as follows: Using an optical arrangement giving a uniform parallel light beam, a FIG. I 2 .

Lt ~ ~ ] ~ ) - } I-ci2

K

L~

t~ ~L 2

HOR.

VERT.

C 24 screen is moved at uniform velocity in front of a photoelectric cell. By cutting out of the screen an aperature to represent the prescribed function, the current in the photocell can be m~ide to have the desired time variation. By the use of thermionic-tube amplifiers, this can be converted to a voltage of suitable magnitude to be applied to the network. Using a number of photo-cells the voltages for successive panel points can be generated successively, with proper overlap in time. Moving the screen in front of the row of photo-cells is then a very analogous procedure to running the train across the bridge. The examination of the structure for different kinds and loading of trains then involves merely the cutting out of a set of screens of proper apertures. The effect of a change in

Mar., I934.] STRUCTURAL ANALYSIS BY ANALOGTES.

32I

speed of t h e train is completely simulated by changing correspondingly the speed at which the screen is moved. A very thorough-going examination of a structure is hence possible. If the mass of the train is comparable with the mass of the structure, it is apparently possible to include its effect, by an extension of the m e t h o d which needs more discussion t h a n can be given in the present m e m o r a n d u m . Now there is another very i m p o r t a n t problem of transient stresses, n a m e l y the problem of the behavior of a structure under earthquake conditions. In this problem, the panel points of the structure which are connected to the foundation are given a forced motion; the nature of which is known roughly at present, and will be known better in the future, as the result of seismographic investigation. Electrically, this problem involves introducing at these panel points a current having a time variation which is in accordance with a prescribed function, obtained as the observed c o m p o n e n t of the velocity of the earth during typical earthquake disturbances. If the complete three dimensional problem is being considered, there are three c o m p o n e n t networks, and into those are introduced the corresponding horizontal, vertical, and transverse components of the motion. This can be done simultaneously, or successively and the results combined by superposition. T h e m e t h o d of producing a current having the desired time variation is very similar to t h a t outlined above for variable forces, with an alteration of the amplifier which will be well understood by those familiar with the use of such devices. T h e m e t h o d of introduction is also clear w i t h o u t great explanation. In Fig. 13 is shown the connection for applying to the structure of Fig. I2 a horizontal motion of the supports of known form. This is done by introducing into the network through transformer T a current whose time variation is a replica of the known motion. Oscillography is used in this case as well, of course, in order to examine stresses and deflections. Now the above t r e a t m e n t of a determinate pin-connected structure under transient conditions applies without any additional features to the problem of the indeterminate pinconnected structure under transient conditions, and hence to a large class of i m p o r t a n t practical problems. The compliance

322

V.

BusH.

[J. F. I.

of members has already been introduced, and when this is properly cared for there is no essential difference between the determinate and indeterminate structures. A further section on the indeterminate pin-connected structure is not necessary. It is hardly necessary to give examples of more complicated structures t h a n are given in the examples, as the more complicated structure simply involves the interconnection of more electrical elements, but no new principles. T h e mathematical t r e a t m e n t of the transient behavior of structures is identical with t h a t of the electrical networks. All of the considerable literature developed by electrical FIG. 13.

/i

12 HOR.

CrvL2

VERT.

fl

engineers is directly applicable to the structural problem. In •considering the characteristics of structures from the point of view of response to transient impressed forces, it is convenient to analyze first for the response to some standard impulse, and then to use the principle of superposition to investigate the response to forces of various time variations. 7 The use of the network in studying the transient behavior of structures m a y hence be limited to an investigation of the response under a 7 A recent treatment of this subject appears in "Calculation of the Stresses Occurrimg in a Building during Earthquakes" by M. Blot, presented before the A. S. M. E. at New Haven, June, I932. See also "Operational Circuit Analysis," loc. cit.

Mar.,

1934.1

STRUCTURAL

ANALYSIS

BY

ANALOGIES.

323

standard impulse. From these one may proceed either by analysis or by further measurement. The major difficulty, that of obtaining the standard response, which m a y readily involve a prohibitive amount of labor from a strictly analytical standpoint, appears to be best obtained by measurement on the electrical analogue. There is an alternative analogy which may sometimes be preferable in studying pin-connected structures under transient conditions. This is obtained by interchanging the r61e of condensers and inductances in the representation above. The analogy then is In the Structure. Mass ....................... Resilience ..................

M S

In the Network. Capacitance... C Inductance .... L

Velocity .................... Displacement ...............

v l

Voltage ....... f E dt

Stress ......................

F

Current

E on a condenser

I through an inductance

The arrangement for the structure of Fig. IO becomes that of Fig. 14 . The static analogy is preserved by connections similar to those of Fig. IO. It is unnecessary to go through the complete treatment. The analogy follows because of the relations.

and

dE

I

dt

C

{. Edt

I = ! d

L

'

which hold for a simple inductance-capacitance circuit. Potential transformers now replace current transformers. Applied loads become applied currents. Velocities of the foundations are represented by applied voltages. There are certain advantages of this arrangement. In general any circuit proposed for purposes of this sort, involving coils and condensers, m a y be reciprocated in this manner. The analogy for the transient case of the frame structure may be built up along similar lines. Unfortunately, it is likely to be seriously complex. The case of the frame of a building, however, admits of a great deal of simplification, since the interesting motions are usually horizontal, and the

[J.

V. BUSH.

324

Fro. x4.

I

3

4

L j2

C

C3

C4

C3

C4

Mar., I934.] STRUCTURAL ANALYSIS BY ANALOGIES.

325

entire load on a floor m a y be considered concentrated. T h e problem thus is reduced to t h a t of determining the elastic restraints a m o n g the various floor levels, of constructing an electric network e m b o d y i n g capacitances to represent the masses, and measuring the response to applied forces. As an example of this procedure, a brief s t u d y was made of the frame structure used for static s t u d y above. The restraints were calculated and are also readily obtainable from the static study. In order to form an estimate of the comparative labor involved, a check was m a d e of static relationships by m e a s u r e m e n t s on a mechanical model. This was built out of drill rod, and the restraints were obtained by m e a s u r e m e n t s of this, using a micrometer eye-piece in a microscope to measure deflections. The various m e t h o d s checked within about five per cent. Undoubtedly, if only a single structure were to be analyzed, the use of a mechanical model offers certain advantages. T h e use of the network for this purpose has its principal advantage when m a n y structures are to be treated. It also gives directly by m e a s u r e m e n t the moments, angles, and shears, which are very hard to measure on a model. The circuit representing the structure is shown in Fig. 15. The equations to be satisfied in the mechanical structure are: F1 = $11D1 + $12D2 --[- $13D~

.F2 = $21D1 + $22D2 -~- S2aDa F3 = $31D1 -4- $32D2 + S~3D3.

The system is considered closed; t h a t is, the reaction of the applied forces appears at the base. T h e D's are deflections; the F's the applied forces. F r o m static measurements of a model, or from m e a s u r e m e n t s of an equivalent electrical network, or from direct numberical computation, a group of deflections corresponding to arbitrary applied forces can be determined. F r o m these m e a s u r e m e n t s the S's (stiffness coefficients) can be calculated. This is not a case of solving for nine unkowns from three equations, as a n u m b e r of indep e n d e n t equations m a y be set up in terms of deflections resulting from a variety of applied forces. As the equations

326

V. BUSH.

[J. F. I.

are written, the terms 311, 322, 333 correspond to the total stiffness in their respective meshes, and not the stiffness in that mesh which is not included in other meshes. Note this difference in terminology from the Cn, C22, C33 of Fig. 15. In setting up the electrical analogue, S is proportional to I/C, and the mass M proportional to L. Note that to supply the term S13 mutual inductance must be relied upon. This is necessary to provide the required isolation of the intermediate mesh. It became apparent, in FIG. 15.

1;1

T1 C12

"1'2

~

T2 C

C2~

' ~

o---

Transient A LI L2 L3

I9.S mh. 19.5 m h . 9.75 mh.

c32

iO----

l o a d i n g of c o n t i n u o u s

B

structure.

C CII I2 2i 22 23 32 33

T"3

IO6 mfd. I2.7 47.3 378 18.2 37,8 558

D Transformer

T1 T2

ratios:

B O : A O = 1.8o B O : A O = 1.45 DC : AO = .386

z s l u g = 6 . 2 7 X 10 -7 h e n r i e s , I ft./lb. = 63S farads, i second = 0.02 second.

solving for the various capacitances, t h a t mutual inductances would be necessary in other places. Otherwise it appears that the sum of the mutual stiffnesses between a given mesh and the remainder of the system is greater than the desired selfstiffness. This would require for its realization the introduction of negative capacitances, which are rather difficult to realize physically, although not impossible.

Mar., I934.]

STRUCTURAL ANALYSIS BY ANALOGIES.

327

Solving for the stiffness coefficients, from the observed deflections, gave the following results: SII 22

33

2.68 X Io 6 lbs./inch. 2.I6

.795

12 1.465 13 - - .268 23 I.OI

Floor loads consistent with the strength of the building were assumed, namely IO6 pounds at the first and second floors, and 0.5 X lO 6 pounds on the top. The transformers used could bear considerable improvem e n t if designed for this purpose. Ideal transformers are needed. T h e approach to this consists of obtaining the greatest possible difference between open and short circuit impedance. T h e difficulty of realizing this is increased by the fact t h a t the transformer m u s t be " i d e a l " over a fairly wide range of frequencies, if it is to be of use in a transient problem. At 60 cycles, which was chosen as the mean of the frequency range considered important, the leakage reactance appeared to be less than five per cent. of the reactance used in the various meshes to represent the masses. The effect of the transformers was minimized by using t h e m as auto transformers where possible, and introducing t h e m between condensers, using ratios approximately equal to the ratio of impedance desired on the two sides. M e a s u r e m e n t s of the elements of the condensers, separately and combined with the transformers, checked within about two per cent. Scales were so chosen t h a t 0.3 to 3.o cycles per second in the mechanical structure corresponded to 15 to 15o c.p.s, in the electrical network. The relative magnitudes of inductance and capacitance were chosen to fit in conveniently with the available transformers. T h e scales, and the magnitudes of the electrical elements, are shown in Fig. 15. An initial transient displacement of the base relative to the rest of the structure corresponds to an initial charge introduced into all meshes. This is accomplished by discharging a condenser from a high potential through the network, from A to D. The potential across the condensers Cn, C2~., Caa will

328

V. BusH.

[J. F. I.

be proportional to the displacements of the respective floors, to the proper scale. Oscillograms of these potentials as a function of time were taken, for the case of an initial impulsive displacement of the base, and for a surge displacement. This latter was obtained by the discharge of a condenser through an inductance and resistance. The curves have been plotted to the same scale, both for time and displacement, for convenience. Nothing has been said so far, about damping. Little is known about it in the usual mechanical structure, and if and when known it can readily be adjusted in the electrical circuit. In the absence of this knowledge, the electrical network contained arbitrary damping and the oscillograms are useful FIG. 16. D I , . S P L A C E M E N T VS.TII'vlE FOR UNIT IMPuLc-IVF...DISPLACEtv/E-NT OF B[i~E.

20

FLOOR ~

o

ZO

TiH¢.

•

H~e.H,

for indicating natural periods, and points of greatest stress for various types of impressed driving forces. Variation of the electrical constants of the network with frequency and current might produce trouble, but it is thought that the values will remain constant within the allowable error, which at the present time is rather large. The system is extremely convenient to handle, in comparison with a mechanical model, and various types of transients can easily be impressed on the system, through the proper sort of driving source. A typical oscillogram is reproduced in Fig. 16. This is presented, not because it indicates anything unusual, but simply as an example of the sort of result obtainable. The networks which have been developed are not the only

Mar., I934.] STRUCTURAL ANALYSIS BY ANALOGIES.

329

ones which may be used. Practical progress in extending our knowledge of the behavior of complex indeterminate structures in the static or transient case by this method will be possible only when special equipment is constructed especially for the purpose, for the conditions which must be met for satisfactory performance are unique. The experimental work here presented, obtained with equipment which was usually not especially adapted to the purpose, is hence preliminary; and it is reported primarily for the purpose of showing the nature of the apparatus necessary and the conditions to be met in its construction. It is believed as the result of this experience, that it is possible to considerably extend, by this method, the facility with which important structures may be analyzed. In this problem, I have been greatly assisted by the criticism and suggestions of Mr. E. L. Rose, who worked out some of the more difficult circuits. Professor H. L. Hazen kindly aided in the transformer design. Mr. W. M. Hall collaborated with me in the experimental portion of the work. All of this help is greatly appreciated. CAMBRIDGE) MASS., August I7, 1933.